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regularly. My first diffidence and constraint have subsided
more and more, and I get accustomed to think more of the
auditors than of myself, and to read in their expressions whether
I should go on or explain the matter further.'
When Dirichlet succeeded Gauss in 1855, Riemann's friends
urged the authorities to appoint Riemann to the security of an
assistant professorship, but the finances of the University could
not be stretched so far. Nevertheless he was granted the equi-
valent of 200 dollars a year, which was better than the uncer-
tainty of half-a-dozen voluntary students' fees. His future
worried him, and when presently he lost both his father and his
sister Clara, making it impossible for hint to escape for vaca-
tions to Quickborn, Riemann felt poor and miserable indeed.
His three remaining sisters went to live with the other brother,
a postal clerk in Bremen whose salary was princely beside that
of the "economically valueless' mathematician.
The following year (1856; Riemann was then thirty) the out-
look brightened a little. It was impossible for a creative genius
like Riemann to be downed by despondency so long as he had
the wherewithal to keep body and soul together in order that
he might work. To this period belong part of his characteristi-
cally original work on Abelian functions, his classic on the
hypergeometric series (see chapter on Gauss) and the differ-
ential equations - of great importance in mathematical physics
- suggested by this series. In both of these works Riemann
struck out on new directions of his own. The generality, the
intuitiveness, of his approach was peculiarly his own. His work
absorbed all his energies and made him happy hi spite of
material worries; possibly, too, the fatal optimism of the con-
sumptive was already at work in him.
Riemann's development of the theory of Abelian functions is
as unlike that of Weierstrass as moonlight is unlike sunlight.
Weierstrass* attack was methodical, exact in all its details, like
the advance of a perfectly disciplined army under a generalship
that foresees everything and provides for all contingencies.
Riemann, for his part, looked over the whole field, seeing
everythmg but the details, which he left to take care of them-
selves, and was content to have grasped the key positions of the